Before the game, Moyes had suggested that victory in North London was essential if they were to make a late surge for the Premier League's top four.
"The games are running out now so it's harder and harder for us to get into the top four," Moyes told Sky Sports News after tonight's match.
"Probably we will get talked about at the end of conversations about the Champions League as we are rank outsiders. But there's no shame in getting a draw here."
The Toffees are now just two points adrift of Chelsea in fourth, but the Blues have two games in hand.